However, invalid formulas (those that are not
entailed by a given theory), cannot always be recognized. In
addition, a consistent formal theory that contains the first-order
theory of the natural numbers (thus having certain "proper
axioms"), by Gödel's
incompleteness theorem, contains true statements which cannot
be proven. In these cases, an automated theorem prover may fail to
terminate while searching for a proof. Despite these theoretical
limits, in practice, theorem provers can solve many hard problems,
even in these undecidable logics.

Related
problems

A simpler, but related, problem is proof
verification, where an existing proof for a theorem is
certified valid. For this, it is generally required that each
individual proof step can be verified by a primitive recursive
function or program, and hence the problem is always
decidable.

Interactive theorem
provers require a human user to give hints to the system.
Depending on the degree of automation, the prover can essentially
be reduced to a proof checker, with the user providing the proof in
a formal way, or significant proof tasks can be performed
automatically. Interactive provers are used for a variety of tasks,
but even fully automatic systems have proven a number of
interesting and hard theorems, including some that have eluded
human mathematicians for a long time.[1][2]
However, these successes are sporadic, and work on hard problems
usually requires a proficient user.

Another distinction is sometimes drawn between theorem proving
and other techniques, where a process is considered to be theorem
proving if it consists of a traditional proof, starting with axioms
and producing new inference steps using rules of inference. Other
techniques would include model checking, which is equivalent to
brute-force enumeration of many possible states (although the
actual implementation of model checkers requires much cleverness,
and does not simply reduce to brute force).

There are hybrid theorem proving systems which use model
checking as an inference rule. There are also programs which were
written to prove a particular theorem, with a (usually informal)
proof that if the program finishes with a certain result, then the
theorem is true. A good example of this was the machine-aided proof
of the four color theorem, which was very
controversial as the first claimed mathematical proof which was
essentially impossible to verify by humans due to the enormous size
of the program's calculation (such proofs are called non-surveyable
proofs). Another example would be the proof that the game Connect Four is a win
for the first player.

Industrial
uses

Commercial use of automated theorem proving is mostly
concentrated in integrated circuit design and verification. Since
the Pentium
FDIV bug, the complicated floating point
units of modern microprocessors have been designed with extra
scrutiny. In the latest processors from AMD, Intel, and others, automated theorem
proving has been used to verify that division and other operations
are correct.

First-order theorem
proving

First-order theorem proving is one of
the most mature subfields of automated theorem proving. The logic
is expressive enough to allow the specification of arbitrary
problems, often in a reasonably natural and intuitive way. On the
other hand, it is still semi-decidable, and a number of sound and
complete calculi have been developed, enabling fully
automated systems. More expressive logics, such as higher order and
modal logics, allow the convenient expression of a wider range of
problems than first order logic, but theorem proving for these
logics is less well developed. The quality of implemented system
has benefited from the existence of a large library of standard
benchmark examples — the Thousands of Problems for Theorem Provers
(TPTP) Problem Library[3] — as
well as from the CADE ATP System Competition
(CASC), a yearly competition of first-order systems for many
important classes of first-order problems.

Some important systems (all have won at least one CASC
competition division) are listed below.

SETHEO is a high-performance system based on the goal-directed
model
elimination calculus. It is developed in the automated
reasoning group of Technical University of
Munich. E and SETHEO have been combined (with other systems) in
the composite theorem prover E-SETHEO.

Waldmeister is a specialized system for unit-equational
first-order logic. It has won the CASC UEQ division for the last
ten years (1997–2006).

Deontic
theorem proving

Deontic logic
concerns normative propositions, such as those used in law,
engineering specifications, and computer programs. In other words,
propositions that are translations of commands or "ought" or "must
(not)" statements in ordinary language. The deontic character of
such logic requires formalism that extends the first-order
predicate calculus. Representative of this is the tool KED.[4]

J. Alan Robinson Syracuse University. Developed original
resolution and unification based first order theorem proving,
co-editor of the "Handbook of Automated Reasoning", recipient of
the Herbrand
Award 1996